Notation for Morphisms

Let $\mathcal{C}$ be a category as in Definition~Category. If $f \in \Hom_{\mathcal{C}}(X,Y)$ we write $$ f : X \to Y. $$ The object $X$ is called the domain of $f$ and $Y$ the codomain. Composition of morphisms is written $$ g \circ f : X \to Z $$ whenever $f : X \to Y$ and $g : Y \to Z$ as in Definition~Category.